Gamma Matrix Generalizationsof the Kitaev Model D. P. Arovas, UCSD

Congjun Wu Zhoushen HuangHsiang-Hsuan Hung

Strongly Interacting Electrons in Low DimensionsPrinceton Center for Theoretical Sciences

September 12, 2011

Why I like these models

1) They remind me of Tinker Toys from my childhood.

2) Geometric pictures and ridiculous Hamiltonians remind me of AKLT models from when I was a postdoc.

3) Many species of Majorana fermions = excuse for figures with pretty colors.

Kitaev honeycomb model (2006):

Kitaev toric code (2003):

magnetic vorticeselectric charges

See A. Kitaev and C. Laumann (2009)

A : gapped abelian phases

N : gapless nonabelian phaseA

A AN

Topologically degenerate ground state (4x on torus) withgap to electric (e) and magnetic (m) excitations that havenontrivial mutual statistics and form composite (e-m) fermions.

Majorana fermions

Self-conjugate (“real”) fermions :

Many species :

2 Majorana = 1 Dirac :

2N Majoranas ⇒ 2N states

S=½ algebra represented with Majoranas

Four Majorana species :

Pauli matrices :

Constraint :

Projector onto constraint sector :

2D Toric CodeApply unitary transformation

to all NW/SE links:

This yields Wen’s model (rotated by π/4)

(construction due to A. Kitaev)

(Wen, 2003)

Define the gauge fields

Magic : Sticks of same color never share a vertexmeans that the gauge fields uij commute!

⇒ classical gauge theory.

Ground state :

H = �JE

X

,E

�xi �z

j �xk�z

l �JM

X

,M

�xi �z

j �xk�z

l

Honeycomb Lattice Model

The honeycomb lattice is threefold coordinated. The magic stick rule still holds,but one Majorana species is free to hop in the presence of a static gauge field.

honeycomb lattice square-octagon lattice(Yang et al. 2007 , Baskaran et al. 2009 , Kells et al. 2010)(Kitaev 2006)

Counting spin, fermion and gauge freedomsConsider N sites on a torus

Spins : degrees of freedom

N Majoranas DOF (

gauge fields DOF

Dirac fermions)DOF

too many!

4 Majoranas per site

Count gauge-invariant freedoms : hexagon fluxes ; 1 constraint

2 Wilson cycles

independent flux variables⇒

Projection onto the constraint sector :

the effect of projection

Suppose . Then but , so

is to sum over all gauge configurations

consistent with a given flux assignment :gauge field WFfermion WF

gauge transformation

sum over gauge configurations

Starting from some we can build the Hilbert space

multiplying by products of even numbers of Majoranas .

preserves (Dirac) fermion number modulo 2.

So everything works out to give :

,

Majorana fermions in a static gauge field

Hamiltonian : with

Problem : must find optimal gauge configuration

Solution : (1) numerical search, (2) Lieb’s theorem (1994)

Time-reversal

Choosing

Time-reversal properties of gauge fields and fluxes :

= # sites on perimeter of p

real, imaginary, with , one has

,

,

even-membered rings ➙ flux is even underodd-membered rings ➙ flux is odd under

Time-reversal broken states : two routes

1) Add explicit T breaking terms :

This yields next-neighbor hopping in the gauge field background.Still noninteracting fermions!

2) Lattices with odd-membered loops

(Kitaev 2006)

Spontaneous breaking of time-reversal when

(Yao and Kivelson 2007)

Ground stateChern number

Lieb’s theoremConsider a general single species Majorana Hamiltonian

on a lattice which has reflection planes which do not intersect any of

.We may assume

gauge field on each link is .

Then the lowest energy assignment

the lattice sites. The

of the plaquette fluxes is one which is reflection symmetric, and one in which

each plaquette p bisected by a reflection plane has flux .

-1

-1

-1

-1

σ

σ

(Lieb, 1994)

Diagonalization of lattice Majorana Hamiltonians

Assume a regular lattice with an even element basis :

where and .

The spectrum consists of the positive eigenvalues of (plus half the zeros).

The spectrum of satisfies

time reversal invariant momenta

The Hamiltonian may be reexpressed in terms of Dirac fermions:

H = iX

k

0Ast(k)

�c†ks ckt �

12 �st

�+

i

4

X

Q

Ast(Q) ⇠s(Q) ⇠t(Q)

Spin-metal in the square-octagon model

with explicit T-breaking terms(Kells et al. 2010)

Yang et al. 2007Baskaran et al. 2009Kells et al. 2010

Lowest energy flux configuration consistent withLieb’s theorem has σ= -1, but adding ring termssuch as to the Hamiltoniancan stabilize the phase with , whichhas a Fermi surface (Baskaran et al. 2009).

Adding next-nearest-neighbor termsexplicitly breaks T and generates new phases,some with exotic Chern numbers (Kells et al. 2010).

Gamma matrices and Clifford algebrasClifford algebra :

When , a representation of the CA can be constructed by tensor

products of Pauli matrices, viz.

In even dimensions, define .

With

Majorana fermion representation

Majorana fermions satisfying .

Then take . The following product is fixed :

Pauli matrices Dirac matrices

Interactions where .

Pauli matrices

Dirac matrices

In addition to and five , ten others :

These form a basis for 4x4 Hermitian matrices.

The construction can be continued for 2kx2k Hermitian matrices :

The symmetry of these various classes under time-reversal must beworked out in detail and depends on conventions for the chargeconjugation operator.

Complex conjugation : one can always takethis is consistent with the constraint

Charge conjugation : for take ; for take

Correspondence to spin tensor algebra for : Murakami et al. 2004Yao et al. 2009Chua et al. 2011

1

5

10 7

5

1

3

rank 0

rank 1

rank 0

rank 2

rank 2

rank 1

rank 3

Models with k=2Consistent with “magic stick rule” we look for a 5-fold coordinated latticeand impose the Hamiltonian

Examples of viable lattices :

Both contain triangular plaquettes and will have T -breaking ground states.

cubic latticeof octahedra

Local projector : ,

Wu et al. 2009Yao et al. 2009

For the decorated square lattice model,the lowest energy flux configuration isthat in which all square plaquettes have

and all triangle fluxes are thesame, with .

Circuit composition rule for flux :

# of common links

edge state structuretopologically nontrivial topologically trivial

ground state : bulk energy bands

Chern number = ±1 Chern number = 0

Spin correlationsThe ground state, when properly projected onto the constraint subspace, is a sumover all gauge configurations consistent with a given flux pattern:Only gauge-invariant objects can have an expectation value. Thus,

Nonabelions forKitaev showed how each vortex binds an odd number of Majorana zero modesin a phase where the Chern number is odd. Yao and Kivelson (2007) observeda degeneracy 2n+1 for n well-separated vortices. We find (at fixed WH/V) n Dirac zero modes for 2n vortices.

vortices and gauge strings

where or . This model has at least twointeresting phases : (i) a gapped chiral spin liquid (C=±2) with abelian vortices, and (ii) a gapless spin liquid with a stable spin Fermi surface (possibly stabilized by

Building on the work of Yao, Zhang, and Kivelson, considered the following model :

additional flux energy terms cf. Baskaran et al. 2007).

(Chua, Yao, and Fiete, 2009)Kagome chiral spin liquid

Diamond lattice modeldiamond = two interpenetrating FCC lattices ,

Our model : k=2 (4x4 Γ matrices). Start with

and add

where

,

This Hamiltonian exhibits deformed Dirac cones at the threeinequivalent X points on the Brillouin zone square faces. Thespectrum is linear in two directions and quadratic in the third.

So far we’ve only used . We now add in corresponding terms where

is replaced by . Then

where and with

primitive RLVs

Here are a gamma-matrix basis for 4x4 Hermitian - not spin operators!

Defining a pseudo-time reversal operator and parity

we consider the most general model which is symmetric under both. This allowsfor the addition of a hybridization term,

This preserves the essential ‘solvability’ of the model in terms of its representationas two Majorana species (0 and 5) hopping in a static Z2 gauge background.

,

.

The Hamiltonian then becomes

where . Pseudo-time reversal symmetry ⇒ :

We can set . Now let J4 ≠ J123 ≡ J, following Fu, Kane, and Mele (2007). Then theDirac nodes at the X points acquire a mass gapproportional to | J4 - J |. The system is thentopologically nontrivial when J4 > J, and thereare an odd number of surface Dirac cones.

Finally, if we relax the requirement of separate and symmetries and requireonly then the band structure is richer, with γ4 , γ14 ,γ45 ,γ24 , and γ34 termsin the Hamiltonian, potentially allowing for a more diverse set of possibilities.

Octahedron cubic lattice model

x_

z_ xz

y

y_

z x

y

x_

y_

z_

z_

z_

z_

1

23

4

5

67

8

A

B1

B2

B3

C1

C2

C3

C4

C5

C6

If we assume a single cube unit cell, thereare ten distinct flux assignments for each octahedron (modulo time reversal).

Several potentially interesting phases appear :

(Z. Huang and DPA, in progress)

However, all these phases violate Lieb’stheorem, which applies here because ofthe presence of reflection planes. Foreach octahedral flux configuration, theenergy is minimized with a reflection-symmetric extension to the full lattice,requiring a 2x2x2 cubic unit cell. TheA phase always has the lowest energy.

A C4

Surface energy plots at :0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5

Ers�E

rs(A

)

⌘/⇡

AB1

B2

B3

C1

C2

C3

C4

C5

C6

k=3 and beyond...Constraint is . Try

If -- not conventional time-reversal !

?

, then

How to represent using algebra?

The 21 and 35 multiplets are of mixed symmetry under the familiar time reversal operation.(I.e. 21=3+5+13 , 35=9+11+15).

Solution : Take , which results in . Then is oddunder time reversal, while are all even :(Y. Li and DPA, 2011)

rank 0

rank 1

rank 2

rank 3

rank 0

rank 1

rank 2

rank 3

rank 4

rank 5

rank 6

rank 7

1

2

5

10

11

15

20

1

7

21

35

3

2

92

2

8

12

3

1244

2421

29

15

11

7

3

1

5

9

1326

2222

18

4

8

2320

4

gammamatrices

spintensors

k=3 Gamma matrix decomposition table(courtesy of Y. Li)

k=3 on the pyrochlore lattice

This generalizes the model of Chua, Yao, andFiete to a k=3 system. The hybridization termKΓ7 may or may not break time reversal.

Once again, there are two Majorana specieshopping in the presence of a single gaugefield, and with on-site hybridization.

Z. Huang (PhD thesis, 2014)

tetrahedra fluxes

hexagon fluxesFixing the fluxes through all the triangular facesof the tetrahedra still leaves the hexagonal facesin all the Kagome planes unconstrained. On asupertetrahedron containing four elementarytetrahedra, there are four hexagonal faces, andwe can flip an even number of them :

minimal unit cell : A2 ground state free hexagon fluxes : A1 ground state

1 + 6 + 1 = 8 hexagon configurations

Total energies for :

fermi surfaces top : bottom :

Selected referencesA. Kitaev, Ann. Phys. 121, 2 (2006)

E. H. Lieb, Phys. Rev. Lett. 73, 2158 (1994) G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett. 98, 247201 (2007) D. H. Lee, G. M. Zhang, and T. Xiang, Phys. Rev. Lett. 99, 196806 (2007) H. Yao and S. A. Kivelson, Phys. Rev. Lett. 99, 247203 (2007) S. Yang, D. L. Zhou, and C. P. Sun, Phys. Rev. B 76, 180404(R) (2007) V. Lahtinen et al., Ann. Phys. 323, 2286 (2008) H. Yao, S.-C. Zhang, and S. A. Kivelson, Phys. Rev. Lett. 102, 217202 (2009) S. Ryu, Phys. Rev. B 79, 075124 (2009) C. Wu, D. P. Arovas, and H. H. Hung, Phys. Rev. B 79, 134427 (2009) K. S. Tikhonov and M. V. Feigel’man, Phys. Rev. Lett. 105, 067207 (2010) V. Chua, H. Yao, and G. Fiete, Phys. Rev. B 83, 180412(R) (2011) G. Kells et al., arXiv 1012.5376